Further Sequences and Series

Sequences and Series

A sequence is a list of terms, while a series is the sum of the terms. A sequence can be defined in two ways:

A recurrence relation, e.g. (for the Fibonacci numbers)
A position-to-term formula, e.g. (for an arithmetic sequence). A sequence with a position-to-term formula can be denoted as , e.g. .

Sequences can be:

Periodic, meaning that there are repeated terms, e.g. . In this example, the period is 3.
Oscillating, meaning they are periodic with just two terms, e.g. .
Convergent, meaning the terms of the sequence get closer to a limiting value, e.g. the terms in converge to 0. A series can also be convergent, meaning the sum to infinity of the sequence has a finite value1. For example, the sum of converges to 2.
Divergent, meaning the terms of the sequence are not convergent, e.g. . A series can also be divergent, meaning the sum to infinity of the sequence does not have a finite value.
Monotonically increasing, meaning each term is greater than the one before it (), e.g. . A monotonically decreasing sequence has each term less than the one before it ().

The sequence of Fibonacci numbers is defined by the recurrence relation , , , giving the first 5 terms as . The ratio of each term to the previous one converges to , the golden ratio. is the positive root of the equation , which is the auxiliary equation for the Fibonacci recurrence (discussed below.)

The sequence of Lucas numbers2 is defined by the same recurrence relation as the Fibonacci numbers (each term is the sum of the two immediately before it), but and , giving the first 5 terms as .

Recurrence relations

A linear recurrence relation is where each term in a sequence is a linear function of previous terms, e.g. . A first-order recurrence relation is where each term only depends on the previous term and some function of , e.g. . For the AS content, only first-order recurrence relations will be considered.

A homogenous recurrence relation has the form , for some constant . In contrast, a non-homogenous recurrence relation has the form , where is some function of (in this course, a polynomial in or ).

A recurrence system is defined by the recurrence relation (e.g. ) and the initial conditions, e.g. . Our goal is typically to find a closed-form solution to the system: a position-to-term rule that only depends on . For the previous example, (you can verify this for ).

Solving recurrence relations

Solving recurrence relations takes a similar method to the method for differential equations with constant coefficients. For a relation of form or , the solution will take the form:

Where is the complementary function and is the particular solution.

First-order recurrence relations

For a first-order recurrence relation, we start by finding the complementary function , which is related to the homogenous part of the recurrence relation:

Start with the reduced equation (only the homogeneous part).
Replace with (), giving the auxiliary equation.
Solve for , giving a complementary function of the form .
In the AS course, is always of the form . Note that if , this will always be a failure case.

Second-order recurrence relations

The method is similar to above but instead we obtain a quadratic in :

Start with the reduced equation (only the homogeneous part).
Replace with (), giving the auxiliary equation.
Solve for , giving two possible roots and .
The complementary function depends on the nature of the roots and :
For distinct real or complex roots and , .
For repeated real roots ,

Particular solution

We then find the particular solution , which is related to the non-homogenous part of the recurrence relation ():

Choose a form for based off :
- If ) is a number,
- If ) is linear,
- If ) is quadratic,
- If ) is an exponential (some constants and ),
A failure case occurs If the form of is the same as for . This requires multiplying by . It is possible for a new failure case to occur, requiring repeated multiplication by until resolved.
Substitute the appropriate form for into both sides of the relation and solve for , , and .

Then, apply the initial conditions to find remaining constants from the complementary function.

Footnotes
1: Footnote on convergence (outside of specification)
There are sequences where the terms converge to 0 that do not sum to a convergent series. For example, the harmonic sequence is convergent because the terms tend to 0. However, the harmonic series is not convergent.
2: Footnote on the Fibonacci and Lucas numbers
is more commonly used to denote the golden ratio than , but the specification uses .
Some definitions of the Fibonacci and Lucas numbers start from instead of (as an extra sidenote, Fibonacci himself started from 1 and 2).